Unexpected biases between congruence classes for parts in k-indivisible partitions

Abstract

For integers k,t≥2, and 1≤r≤t let Dk×(r,t;n) be the number of parts among all k-indivisible partitions of n (i.e., partitions where all parts are not divisible by k) of n that are congruent to r modulo t. Using Wright's circle method, we derive an asymptotic for Dk×(r,t;n) as n→∞ when k,t are coprime. The main term of this asymptotic does not depend on r, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo t. This induces an ordering on the congruence classes modulo t, which we call the k-indivisible ordering. We prove that for k≥6(t2−1)π2 the k-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k<6(t2−1)π2.